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Theorem ixxss1 13390

Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)

**Hypotheses**
Ref	Expression
ixx.1	⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
ixxss1.2	⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)})
ixxss1.3	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤))

**Assertion**
Ref	Expression
ixxss1	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶))

Distinct variable groups: 𝑥,𝑤,𝑦,𝑧,𝐴 𝑤,𝐶,𝑥,𝑦,𝑧 𝑤,𝑂 𝑤,𝐵,𝑥,𝑦,𝑧 𝑤,𝑃 𝑥,𝑅,𝑦,𝑧 𝑥,𝑆,𝑦,𝑧 𝑥,𝑇,𝑦,𝑧 𝑤,𝑊 Allowed substitution hints: 𝑃(𝑥,𝑦,𝑧) 𝑅(𝑤) 𝑆(𝑤) 𝑇(𝑤) 𝑂(𝑥,𝑦,𝑧) 𝑊(𝑥,𝑦,𝑧)

**Proof of Theorem ixxss1**
Step	Hyp	Ref	Expression
1		ixxss1.2	. . . . . . . 8 ⊢ 𝑃 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑇𝑧 ∧ 𝑧𝑆𝑦)})
2	1	elixx3g 13385	. . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) ↔ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) ∧ (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶)))
3	2	simplbi 496	. . . . . 6 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
4	3	adantl 480	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*))
5	4	simp3d 1141	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ ℝ^*)
6		simplr 767	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑊𝐵)
7	2	simprbi 495	. . . . . . 7 ⊢ (𝑤 ∈ (𝐵𝑃𝐶) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶))
8	7	adantl 480	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝐵𝑇𝑤 ∧ 𝑤𝑆𝐶))
9	8	simpld 493	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵𝑇𝑤)
10		simpll 765	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴 ∈ ℝ^*)
11	4	simp1d 1139	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐵 ∈ ℝ^*)
12		ixxss1.3	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝑤 ∈ ℝ^*) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤))
13	10, 11, 5, 12	syl3anc 1368	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → ((𝐴𝑊𝐵 ∧ 𝐵𝑇𝑤) → 𝐴𝑅𝑤))
14	6, 9, 13	mp2and 697	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐴𝑅𝑤)
15	8	simprd 494	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤𝑆𝐶)
16	4	simp2d 1140	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝐶 ∈ ℝ^*)
17		ixx.1	. . . . . 6 ⊢ 𝑂 = (𝑥 ∈ ℝ^, 𝑦 ∈ ℝ^ ↦ {𝑧 ∈ ℝ^* ∣ (𝑥𝑅𝑧 ∧ 𝑧𝑆𝑦)})
18	17	elixx1 13381	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^ ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
19	10, 16, 18	syl2anc 582	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → (𝑤 ∈ (𝐴𝑂𝐶) ↔ (𝑤 ∈ ℝ^* ∧ 𝐴𝑅𝑤 ∧ 𝑤𝑆𝐶)))
20	5, 14, 15, 19	mpbir3and 1339	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) ∧ 𝑤 ∈ (𝐵𝑃𝐶)) → 𝑤 ∈ (𝐴𝑂𝐶))
21	20	ex 411	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) → (𝑤 ∈ (𝐵𝑃𝐶) → 𝑤 ∈ (𝐴𝑂𝐶)))
22	21	ssrdv 3984	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴𝑊𝐵) → (𝐵𝑃𝐶) ⊆ (𝐴𝑂𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 {crab 3419 ⊆ wss 3946 class class class wbr 5145 (class class class)co 7416 ∈ cmpo 7418 ℝ^*cxr 11288

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7995 df-2nd 7996 df-xr 11293

This theorem is referenced by: iooss1 13407 limsupgord 15469 pnfnei 23212 dvfsumrlimge0 26053 dvfsumrlim2 26055 tanord1 26561 rlimcnp 26990 rlimcnp2 26991 dchrisum0lem2a 27543 pntleml 27637 pnt 27640 liminfgord 45411

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